ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
BLOW-UP CRITERIA AND INSTABILITY OF STANDING WAVES FOR THE INHOMOGENEOUS FRACTIONAL
SCHR ¨ODINGER EQUATION
BINHUA FENG, ZHIQIAN HE, JIAYIN LIU
Abstract. In this article, we study the blow-up and instability of standing waves for the inhomogeneous fractional Schr¨odinger equation
i∂tu−(−∆)su+|x|−b|u|pu= 0,
wheres∈(12,1), 0< b <min{2s, N}and 0< p < 4s−2bN−2s. In theL2-critical andL2-supercritical cases, i.e.,4s−2bN ≤p < 4s−2bN−2s, we establish general blow- up criteria for non-radial solutions by using localized virial estimates. Based on these blow-up criteria, we prove the strong instability of standing waves.
1. Introduction
Over the past decade, there has been a great deal of interest in studying the fractional Schr¨odinger equation
i∂tu= (−∆)su+f(u), (1.1)
where 0 < s < 1 and f(u) is the nonlinearity. The fractional differential oper- ator (−∆)s is defined by (−∆)su = F−1[|ξ|2sF(u)], where F and F−1 are the Fourier transform and inverse Fourier transform, respectively. Equation (1.1) was first deduced by Laskin in [24, 25] by extending the Feynman path integral from the Brownian-like to the L´evy-like quantum mechanical paths. The fractional Schr¨odinger equation also arises in the description of Bonson stars as well as in water wave dynamics (see [16]) and in the continuum limit of discrete models with long-range interactions (see [23]).
In this article, we consider the blow-up criteria and instability of standing waves for the inhomogeneous fractional Schr¨odinger equation
i∂tu−(−∆)su+|x|−b|u|pu= 0, (t, x)∈[0, T∗)×RN,
u(0, x) =u0(x), (1.2)
where u : [0, T∗)×RN → C is the complex valued function, N ≥ 1, u0 ∈ Hs, 0< s <1, 0< b <min{2s, N}, 0< p < 4s−2bN−2s.
2010Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92, 92C17.
Key words and phrases. Inhomogeneous fractional Schr¨odinger equation; blow-up criteria;
strong instability.
c
2021 Texas State University.
Submitted November 28, 2020. Published May 7, 2021.
1
This equation enjoys the scaling invariance. That is, if u(t, x) is a solution of (1.2), then
uλ(t, x) =λ2s−bp u(λ2st, λx) for allλ >0, is also a solution of (1.2). By simple calculations, we have
kuλ(t)kH˙s=λs+2s−bp −N2ku(λ2st)kH˙s. Thus, the critical Sobolev index is given by
sc:= N
2 −2s−b
p . (1.3)
Whensc<0, equation (1.2) isL2-subcritical. The smallest power for which blow- up may occur isp= 4s−2bN , which is referred to L2-critical case corresponding to sc = 0. When 0 < sc < s, (1.2) is L2-supercritical and Hs-subcritical. When sc =s, (1.2) isHs-critical. In this paper, we are interested in the L2-critical and L2-supercritical cases. Therefore, we restrict our attention to the case 0≤sc < s.
Rewriting this condition in terms ofp, we obtain 4s−2b
N ≤p < 4s−2b N−2s.
If one considers initial data in Hs, then the equation enjoys mass and energy conservation laws:
M(u(t)) :=ku(t)kL2 =ku0kL2, (1.4) and
E(u(t)) :=1 2
Z
RN
|(−∆)s/2u(t, x)|2dx− 1 p+ 2
Z
RN
|x|−b|u(t, x)|p+2dx
=E(u0).
(1.5) Before entering some details of our results, let us recall known blow-up results . For the classical Shr¨odinger equation, i.e., s= 1, the Variance-Virial Law holds, that is
1 2
d dt
Z
RN
|x|2|u(t, x)|2dx= 2 Im Z
RN
¯
u(t, x)x· ∇u(t, x)dx, (1.6) provided initial data u0 ∈ Σ := {u0 ∈ H1 and xu0 ∈ L2}. Combining (1.6) and the virial identity, one can obtain blow-up results for the classical Schr¨odinger equation with negative energy E(u0) <0 and finite variance, see [2]. Ogawa and Tsutsumi [26] removed the assumptionu0∈Σ for the radial symmetry initial data.
Applying similar ideas, whens= 1, Farah [10] and Dinh [7] established the blow-up criteria for equation (1.2) with initial data u0 ∈Σ :={v ∈H1 and xv∈L2} and radial symmetry initial data. However, when s <1, identity (1.6) fails and these arguments cannot work. However, a generalization of the variance for the fractional Schr¨odinger equation is given by
V(s)[u(t)] :=
Z
RN
¯
u(t, x)x·(−∆)1−sxu(t, x)dx=kx(−∆)1−s2 u(t)k2L2. (1.7) Letu(t) be the solution of equationi∂tu= (−∆)su, a formal calculation yields
1 2
d
dtV(s)[u(t)] := 2 Im Z
RN
¯
u(t, x)x· ∇u(t, x)dx. (1.8) Based on this identity, the authors in [3, 4, 33] successfully obtained blow-up results for (1.1) with radial initial data and Hartree-type nonlinearity, i.e.,f(u) =−(|x|−γ∗
|u|2)uwithγ≥1. Because it is very hard to control the nontrivial error terms, this method fails to work for the local nonlinearitiesf(u) =−|u|pu, see [1]. Using the Balakrishman’s formula
(−∆)s= sinπs π
Z ∞
0
ms−1 −∆
−∆ +mdm, (1.9)
Boulenger, Himmelsbach and Lenzmann [1] established the differential estimate d
dt
Im Z
RN
u(t)∇ϕ¯ R· ∇u(t)dx
≤4pN E(u0)−2δk(−∆)s/2u(t)k2L2+◦R(1)(1 +k(−∆)s/2u(t)kp/s+L2 ), where δ=pN−2s. Based on this key estimate and a standard comparison ODE argument, they proved the existence of radial blow-upHssolutions.
For the inhomogeneous fractional Schr¨odinger equation (1.2), Peng and Zhao in [27] obtained the existence of radial blow-up solutions. In this paper, by using localized virial estimates and the ideas of Du, Wu and Zhang [9], we remove this assumption, and establish general blow-up criteria for non-radial solutions in the L2-critical andL2-supercritical cases. The main difficulty is the appearance of the fractional order Laplacian (−∆)s and the singular potential|x|−b. Whens= 1, it easily follows that the time derivative of the virial action
1 2
d dt
Z
RN
ϕ(x)|u(t, x)|2dx= 2 Im Z
RN
u(t, x)∇ϕ(x)¯ · ∇u(t, x)dx. (1.10) By applying this identity, Du, Wu and Zhang [9] established anL2-estimate in the exterior ball. Based on thisL2-estimate and the virial estimates, they established blow-up criteria for the classical Schr¨odinger equation. In the case s ∈ (12,1), the identity (1.10) does not hold. However, by using the Balakrishman’s formula (1.9) and exploiting the ideas in [1], we can obtain the time derivative of the virial action. We consequently obtain the following general blow-up criteria for non-radial solutions in bothL2-critical andL2-supercritical cases.
Theorem 1.1. Let N ≥1, s∈(12,1), 4s−2bN ≤p < 4s−2bN−2s, and u0∈Hs. Assume that u∈C([0, T∗), Hs)is a solution of (1.2). Furthermore, we assume that either E(u0)<0, or, ifE(u0)≥0 and
E(u0)scku0k2(s−sL2 c)< E(Q)sckQk2(s−sL2 c), k(−∆)s/2u0ksLc2ku0ks−sL2 c>k(−∆)s/2QksLc2kQks−sL2 c,
. (1.11)
wheresc is defined by (1.3)andQis the ground state of the elliptic equation (−∆)sQ+Q− |x|−b|Q|pQ= 0. (1.12) Then one of the following statements holds:
• u(t)blows up in finite time, i.e. T∗<+∞;
• u(t) blows up infinite time, i.e., there exists (tn)n≥1 such that tn →+∞
and
n→∞lim k(−∆)s/2u(tn)kL2=∞.
Our blow-up criteria also hold for (1.2) with s= 1, which to our knowledge is new. Whens= 1, similar blow-up criteria for (1.2) with radial solutions or initial data u0 ∈Σ :={v ∈H1 andxv∈L2} have been established in [7, 10]. Here, we remove the assumption of radial solutions and u0 ∈Σ :={v ∈H1 andxv∈L2}.
So our results improve some previous results. Based on blow-up criterion (1.11), we can prove the strong instability of standing waves of (1.2).
Firstly, we introduce some notation. Throughout this paper, we call a standing wave solution of (1.2) of the formeiωtQω, whereω∈Ris a frequency andQω∈Hs is a nontrivial solution to the elliptic equation
(−∆)sQω+ωQω− |x|−b|Qω|pQω= 0. (1.13) LetQω(x) =ω2s−b2sp Q(ω2s1x) in (1.13), thenQsatisfies equation (1.12). In particular, by some basic calculations, we have
E(Qω)sckQωk2(s−sL2 c)=E(Q)sckQk2(s−sL2 c), (1.14) k(−∆)s/2QωksLc2kQωks−sL2 c =k(−∆)s/2QksLc2kQks−sL2 c. (1.15) In fact, these two quantities are scaling invariant of (1.2).
Definition 1.2. A functionQ∈Hs\{0}is called a ground state for (1.12) if it is a minimizer of the Weinstein’s functional
J(v) :=kvk
N p+2b 2s
H˙s kvkp+2−
N p+2b 2s
L2
R
RN|x|−b|v(x)|p+2dx , (1.16) that is,
J(Q) = inf{J(v) :v∈Hs\{0}}. (1.17) The existence of ground states related to (1.12) has been established in Lemma 2.2. In addition, a direct computation shows that
kQωkL2 =ω4s−2b−N p4sp kQkL2, kQωkH˙s=ω2sp−N p+4s−2b 4sp kQkH˙s, Z
RN
|x|−b|Qω(x)|p+2dx=ω2sp−N p+4s−2b 2sp
Z
RN
|x|−b|Q(x)|p+2dx.
These imply that
J(Qω) =J(Q).
That is,Qωis also a minimizer of the Weinstein’s functional. Thus, we can define the ground states related to (1.13) as follows: A functionQω ∈ Hs\{0} is called a ground state solution of (1.13) if it is a minimizer of the Weinstein’s functional (1.16). We can derive ground states of (1.13) from ground states related to (1.12).
This implies the existence of ground states related to (1.13) when ω > 0. In addition, the uniqueness of ground states related to (1.12) is an open problem.
Note also that (1.13) can be written asSω0(Qω) = 0, where Sω(Q) :=E(Q) +ω
2kQk2L2
=1
2kQk2H˙s+ω
2kQk2L2− 1 p+ 2
Z
RN
|x|−b|Q(x)|p+2dx,
(1.18) is the action functional. We also define the following functional
K(Q) :=∂λSω(Qλ)|λ=1=skQk2H˙s−N p+ 2b 2p+ 4
Z
RN
|x|−b|Q(x)|p+2dx, (1.19) where
Qλ(x) :=λN/2Q(λx). (1.20)
To the best of our knowledge, the general method to investigate the strong in- stability of standing waves for the classical Schr¨odinger equation is to apply the
variational characterization of the ground states as minimizers of the action func- tional and derive the key estimateK(u(t))≤2(Sω(u0)−Sω(Qω)). Then, it follows from the virial identity that
d2
dt2kxu(t)k2L2 = 8K(u(t))≤16(Sω(u0)−Sω(Qω)),
where K(u(t)) is defined by (1.19) withs= 1. Finally, one can choose the initial datau0such thatSω(u0)−Sω(Qω)<0. This implies that the solutionu(t) of (1.1) with s= 1 blows up in finite time. Thus, one can prove the strong instability of ground state standing waves, see [2, 6, 12, 13, 14, 15, 22, 29, 30].
Here, we present a simpler method to study the strong instability of standing waves, which is based on the blow-up criterion (1.11).
Theorem 1.3. LetN ≥1,s∈(12,1), 4s−2bN ≤p < 4s−2bN−2s,ω >0,Qω be the ground state related to (1.13). Then, the standing wave u(t, x) = eiωtQω(x) is strongly unstable in the following sense: there exists {u0,n} ⊂ Hs such thatu0,n →Qω in Hs as n → ∞ and the corresponding solution un of (1.2) with initial data u0,n
blows up in finite or infinite time for anyn≥1.
In previous results, to construct blow-up solutions around the ground state solution, one needs to assume that the ground state solution Qω is radial or Qω ∈ Σ :={v ∈ H1andxv ∈ L2}. Here, we remove these assumptions, so our result greatly improve some previous results.
This article is organized as follows: in Section 2, we recall and prove some lemmas such as the local well-posedness theory of (1.2), Brezis-Lieb’s lemma, the sharp Gagliardo-Nirenberg type inequality (2.1) and the localized virial estimate related to (1.2). In section 3, we establish blow-up criteria for (1.2). In section 4, we prove the strong instability of standing waves.
Throughout this article, we use the following notation. C > 0 stands for a constant that may be different from line to line when it does not cause any confusion.
For anys∈(0,1), the fractional Sobolev spaceHs(RN) is defined by Hs(RN) =
u∈L2(RN) : Z
RN
(1 +|ξ|2s)|ˆu(ξ)|2dξ <∞ , endowed with the norm
kukHs(RN)=kukL2(RN)+kukH˙s(RN), where up to a multiplicative constant,
kukH˙s(RN)=nZZ
RN×RN
|u(x)−u(y)|2
|x−y|N+2α dx dyo1/2
is the so-called Gagliardo semi-norm ofu. In this paper, we often use the abbrevi- ationsLr=Lr(RN),Hs=Hs(RN).
2. Preliminary lemmas
In this section, we recall some preliminary results that will be used later. Firstly, let us recall the local theory for the Cauchy problem (1.2). By applying Strichartz’s estimates and the contraction mapping argument, Hong and Sire in [21] first studied the local well-posedness for the fractional Schr¨odinger equation in Hs. Because Strichartz’s estimates have a loss of derivatives in the non-radial symmetry case, a weak local well-posedness follows in the energy space compared to the classical
Schr¨odinger equation, see [5, 21] for more details. In the radial symmetry case, one can remove the loss of derivatives in Strichartz’s estimates. But it needs a restriction on the validity ofs, namely 2NN−1 ≤s <1.
For the inhomogeneous Schr¨odinger equation (1.2) with s = 1, Genoud and Stuart [17] first studied the well-posedness by using the argument of Cazenave [2]. By using Strichartz’s estimates and the contraction mapping argument, Guz- man [19] also established the local well-posedness as well as the small data global well-posedness in Sobolev spaces. By using radial Strichartz’s estimates and the contraction mapping argument, we can obtain the following local well-posedness for (1.2) with radialHs initial data. The proof is standard, see [5, 19, 21]. So we omit it.
Theorem 2.1. LetN ≥2, 2N−1N ≤s <1,0< p < 4s−2bN−2s and0< b <min{2s, N}.
If u0 ∈ Hs is radial, then there exists T = T(ku0kHs) such that (1.2) admits a unique solution u ∈ C([0, T], Hs). Let [0, T∗) be the maximal time interval on which the solution u is well-defined, if T∗ < ∞, then ku(t)kH˙s → ∞ as t ↑ T∗. Moreover, for all 0≤t < T∗, the solution u(t)satisfies the conservations of mass and energy.
Next, we recall the following sharp Gagliardo-Nirenberg inequality, which has been established in [27].
Lemma 2.2 ([27]). Let 0< s <1,0< p < 4s−2bN−2s and0< b <min{2s, N}. Then, for allu∈Hs,
Z
RN
|x|−b|u(x)|p+2dx≤Coptkuk
N p+2b 2s
H˙s kukp+2−
N p+2b 2s
L2 , (2.1)
where the best constantCopt is given by
Copt= N p+ 2b 2s(p+ 2)−(N p+ 2b)
4s−(N p+2b)
4s 2s(p+ 2)
(N p+ 2b)kQkpL2
,
whereQis the ground state of (1.12). Moreover, the following Pohozaev’s identities hold
kQk2H˙s= N p+ 2b 2s(p+ 2)
Z
RN
|x|−b|Q|p+2dx= N p+ 2b
2s(p+ 2)−(N p+ 2b)kQk2L2. (2.2) Lemma 2.3 ([1]). Let N ≥1,ϕ:RN →R and∇ϕ∈W1,∞(RN). Then, for all u∈H1/2, it follows that
Z
RN
u(x)∇ϕ(x)· ∇u(x)dx
≤Ck∇ϕkW1,∞
k|∇|1/2uk2L2+kukL2k|∇|1/2ukL2
,
whereC >0 depends only onN.
To study localized virial estimates for (1.2), we introduce an auxiliary function um(x) :=cs 1
−∆ +mu(x) =csF−1 u(ξ)ˆ
|ξ|2+m
, m >0, (2.3) wherecs:=p
sin(πs)/π.
Lemma 2.4 ([1]). Let N ≥ 1, s ∈ (0,1), ϕ : RN → R and ∆ϕ ∈ W2,∞(RN).
Then, for all u∈L2,
Z ∞
0
ms Z
RN
(∆2ϕ)|um|2dx dm
≤Ck∆2ϕksL∞k∆ϕk1−sL∞kuk2L2,
whereC >0 depends only ons andN.
Applying the identity sinπs
π Z ∞
0
ms
(|ξ|2+m)2dm=s|ξ|2s−2, we deduce form the Plancherel’s and Fubini’s theorems that
Z ∞
0
ms Z
RN
|∇um|2dx dm= Z
RN
sinπs π
Z ∞
0
msdm (|ξ|2+m)2
|ξ|2|ˆu(ξ)|2dξ
= Z
RN
(s|ξ|2s−2)|ξ|2|u(ξ)|ˆ 2dξ=sk(−∆)s/2uk2L2, (2.4)
for anyu∈H˙s.
Lemma 2.5 ([8]). Let N ≥1, s∈(1/2,1),ϕ: RN →R and∇ϕ∈W1,∞. Then for any u∈L2,
Z ∞
0
ms Z
RN
(∆ϕ)|um|2dx dm
≤Ck∆ϕk2s−1L∞ k∇ϕk2−2sL∞ kuk2L2, whereC >0 depends only ons andN.
Lemma 2.6 ([8]). Let N ≥1, s∈(1/2,1),ϕ: RN →R and∇ϕ∈W1,∞. Then for any u∈H1/2,
Z ∞
0
ms Z
RN
um∇ϕ· ∇umdx dm
≤Ck∇ϕkW1,∞kuk2H1/2, whereC >0 depends only onN.
Lemma 2.7 (Virial identity). Let N ≥1, s ∈ (1/2,1) and ϕ :RN → R be such that ϕ∈W2,∞. Assume that u∈C([0, T∗), Hs)is a solution to (1.2). Then
d
dtVϕ[u(t)] =−i Z ∞
0
ms Z
RN
(∆ϕ)|um(t)|2dx dm
−2i Z ∞
0
ms Z
RN
um(t)∇ϕ· ∇um(t)dx dm
(2.5)
for any t∈[0, T∗), where
Vϕ[u(t)] :=
Z
RN
ϕ(x)|u(t, x)|2dx
is the localized virial action ofuassociated to ϕandum(t) =cs(−∆ +m)−1u(t).
Proof. Because the general case follows by an approximation argument, we only prove (2.5) foru∈C0∞(RN). Sinceu(t) satisfies (1.2), it easily follows that
d
dtVϕ[u(t)] = d
dthu(t), ϕu(t)i=ihu(t),[(−∆)s, ϕ]u(t)i,
where [X, Y] =XY −Y X is the commutator ofX andY. To study [(−∆)s, ϕ], we use the fact that for operatorsA≥0,B andm >0 any positive real number,
A A+m, B
=
1− m
A+m, B
=−m 1
A+m, B
=m 1
A+m[A, B] 1 A+m, see [1]. Using this identity with A = (−∆)s and B = ϕ, by the Balakrishman’s formula we have
[(−∆)s, ϕ] = sinπs π
Z ∞
0
ms −∆
−∆ +m, ϕ dm
= sinπs π
Z ∞
0
ms 1
−∆ +m[−∆, ϕ] 1
−∆ +mdm.
Thus,
hu(t),[(−∆)s, ϕ]u(t)i
=
u(t),sinπs π
Z ∞
0
ms 1
−∆ +m[−∆, ϕ] 1
−∆ +mdm u(t)
=c2s Z ∞
0
mshu(t), 1
−∆ +m[−∆, ϕ] 1
−∆ +mu(t)idm
= Z ∞
0
mshcs(−∆ +m)−1u(t),[−∆, ϕ]cs(−∆ +m)−1u(t)idm
= Z ∞
0
ms Z
RN
um(t) (−∆ϕum(t)−2∇ϕ· ∇um(t))dx dm
= Z ∞
0
ms Z
RN
(−∆ϕ)|um(t)|2−2um(t)∇ϕ· ∇um(t) dx dm.
The proof is complete.
The following estimate is a direct consequence of Lemmas 2.5, 2.6 and 2.7.
Corollary 2.8. Let N ≥1, s∈(1/2,1) andϕ:RN →Rbe such that ϕ∈W2,∞. Assume thatu∈C([0, T∗), Hs)is a solution to (1.2). Then for anyt∈[0, T∗),
|d
dtVϕ[u(t)]| ≤Ck∇ϕkW1,∞ku(t)k2Hs, for some constant C >0 depending only ons andN.
Now we define the localized Morawetz action ofuassociated toϕby Mϕ[u(t)] := 2 Im
Z
RN
u(t, x)∇ϕ(x)¯ · ∇u(t, x)dx. (2.6) By Lemma 2.3, we obtain the bound
|Mϕ[u(t)]| ≤C k∇ϕkL∞,k∆ϕkL∞
ku(t)k2H1/2.
Hence the quantityMϕ[u(t)] is well-defined, sinceu(t)∈Hswith somes >1/2 by assumption. By a similar argument as that in [1, Lemma 2.1], we have the following time evolution ofMϕ[u(t)].
Lemma 2.9(Morawetz identity). LetN ≥1, s∈(1/2,1)andϕ:RN →Rbe such that ∇ϕ∈W3,∞. Assume thatu∈C([0, T∗), Hs)is a solution to (1.2). Then for any t∈[0, T∗), it holds that
d
dtMϕ[u(t)]
= Z ∞
0
ms Z
RN
n
4∂kum(t)(∂kl2ϕ)∂lum(t)−(∆2ϕ)|um(t)|2o dx dm
− 2p p+ 2
Z
RN
∆ϕ|x|−b|u(t, x)|p+2dx
− 4b p+ 2
Z
RN
|x|−b−2x· ∇ϕ|u(t, x)|p+2dx,
(2.7)
whereum(t) =um(t, x) is defined by (2.3).
Proof. It follows from an integration by parts that hu(t),[−|x|−b|u(t)|p, iΓϕ]u(t)i
=−hu(t),[|x|−b|u(t)|p,∇ϕ· ∇+∇ · ∇ϕ]u(t)i
= 2 Z
RN
|x|−b|u(t, x)|2∇ϕ· ∇|u(t, x)|pdx+ 2 Z
RN
|u(t, x)|p+2∇ϕ· ∇|x|−bdx
=− 2p p+ 2
Z
RN
∆ϕ|x|−b|u(t, x)|p+2dx− 4b p+ 2
Z
RN
|x|−b−2x· ∇ϕ|u(t, x)|p+2dx, where we used the identities
∇|x|−b=−b|x|−b−2x and ∇|u|p+2=p+ 2
p ∇|u|p|u|2.
Following the method used in [1], we complete the proof.
3. Blow-up criteria
In this section, we will prove Theorem 1.1. Firstly, we establish the following blow-up criteria for (1.2).
Lemma 3.1. Let N≥1,s∈(12,1), 4s−2bN ≤p < 4s−2bN−2s. Assume thatu0∈Hsand u∈C([0, T∗), Hs)is the corresponding solution of (1.2). If there exists δ >0such that
K(u(t))≤ −δ (3.1)
for allt∈[0, T∗), then one of the following two statements holds:
• u(t)blows up in finite time, i.e.T∗<+∞;
• u(t)blows up infinite time and there exists(tn)n≥1such thattn→+∞and
n→∞lim k(−∆)s/2u(tn)kL2 =∞. (3.2) Proof. If T∗ < +∞, then the proof is done. If T∗ = +∞, we prove (1.1) by contradiction. If not, the solutionu(t) exists globally and there existsC0>0 such that
C0:= sup
t∈[0,+∞)
k(−∆)s/2u(t)kL2 <∞. (3.3) This, together with the conservation of mass, implies that
C1:= sup
t∈[0,+∞)
ku(t)kHs <∞. (3.4) Now, we claim that for everyη >0,R >1, there exists a constantC >0 indepen- dent ofRandC1 such that for anyt∈[0,CCηR2
1
], Z
|x|≥R
|u(t, x)|2dx≤η+oR(1). (3.5) To this end, we define a smooth functionθ: [0,∞)→[0,1] that satisfies
θ(r) =
(0 if 0≤r≤1/2, 1 ifr≥1.
ForR >1, we define the radial function
φR(x) =φR(r) :=θ(r/R), r=|x|.
It easily follows that
∇φR(x) = x
rRθ0(r/R), ∆φR(x) = 1
R2θ00(r/R) +(N−1)
rR θ0(r/R).
In particular, we have
k∇φRkW1,∞ ∼ k∇φRkL∞+k∆φRkL∞ ≤CR−1. (3.6) Now, we can define the localized virial potential
VφR[u(t)] :=
Z
RN
φR(x)|u(t, x)|2dx.
We have
VφR[u(t)] =VφR[u0] + Z t
0
d
dτVφR[u(τ)]dτ
≤ VφR[u0] + sup
τ∈[0,t]
d
dτVφR[u(τ)]
t.
By Corollary 2.8, (3.4) and (3.6), we obtain sup
τ∈[0,t]
d
dτVφR[u(τ)]
≤Ck∇φRkW1,∞ sup
τ∈[0,t]
ku(τ)k2Hs ≤CC12R−1, for some constantC >0 independent ofR andC1. We thus obtain
VφR[u(t)]≤ VφR[u0] +CC12R−1t,
for allt≥0. By the choice ofθand the conservation of mass, we have VφR[u0] =
Z
RN
φR(x)|u0(x)|2dx≤ Z
|x|>R/2
|u0(x)|2dx→0, asR→ ∞orVφR[u0] =oR(1). On the other hand, we have
Z
|x|≥R
|u(t, x)|2dx≤ VφR[u(t)].
Collecting the above estimates, we can obtain the control on theL2-norm of the solution outside a large ball, i.e., claim (3.5).
Next, we assume thatϕ(x) =ϕ(r) is radial and satisfies ϕ(r) =
(r2/2 forr≤1, const. forr≥10,
andϕ00(r)≤1 forr≥0. GivenR >0 , we define the rescaled functionϕR:RN →R by
ϕR(x) :=R2ϕ x R
. (3.7)
We readily verify the inequalities
1−ϕ00R(r)≥0, 1−ϕ0R(r)
r ≥0, N−∆ϕR(x)≥0, for allr≥0 and allx∈RN. It is easy to see that
k∇kϕRkL∞ ≤CR2−k, k= 0,· · ·,4, and
supp(∇kϕR)⊂
({x:|x| ≤10R} fork= 1,2, {x:R≤ |x| ≤10R} fork= 3,4.
By Lemma 2.9, we have d
dtMϕR[u(t)]
= Z ∞
0
ms Z
RN
n
4∂kum(t)(∂kl2ϕR)∂lum(t)−(∆2ϕR)|um(t)|2o dx dm
− 2p p+ 2
Z
RN
∆ϕR|x|−b|u(t, x)|p+2dx
− 4b p+ 2
Z
RN
|x|−b−2x· ∇ϕR|u(t, x)|p+2dx
(3.8)
where um(t) = um(t, x) is defined in (2.3). Since supp(∆2ϕR) ⊂ {|x| ≥ R}, by Lemma 2.4, we have
Z ∞
0
ms Z
RN
(∆2ϕR)|um(t)|2dx dm
≤Ck∆2ϕRksL∞k∆ϕRk1−sL∞ku(t)k2L2(|x|≥R)
≤CR−2sku(t)k2L2(|x|≥R).
(3.9) SinceϕR is radial, we use
∂jk2 =δjk
r −xjxk
r3
∂r+xjxk
r2 ∂r2 to write
Z ∞
0
ms Z
RN
∂kum(t)(∂jk2 ϕR)∂lum(t)dx dm
= Z ∞
0
ms Z
RN
ϕ0R
r |∇um(t)|2dx dm +
Z ∞
0
ms Z
RN
ϕ00R r2 −ϕ0R
r3
|x· ∇um(t)|2dx dm.
Using (2.4), we write Z ∞
0
ms Z
RN
ϕ0R
r |∇um(t)|2dx dm
=sk(−∆)s/2u(t)k2L2+ Z ∞
0
ms Z
RN
ϕ0R r −1
|∇um(t)|2dx dm.
Sinceϕ00R≤1, the Cauchy-Schwarz inequality implies Z ∞
0
ms Z
RN
ϕ0R r −1
|∇um(t)|2dx dm +
Z ∞
0
ms Z
RN
ϕ00R−ϕ0R r
|x· ∇um(t)|2
r2 dx dm≤0.
Therefore, 4
Z ∞
0
ms Z
RN
∂kum(t)(∂jk2 ϕR)∂lum(t)dx dm≤4sk(−∆)s/2u(t)k2L2. (3.10)
We next write
− 2p p+ 2
Z
RN
|x|−b|u(t, x)|p+24ϕRdx
=−2pN p+ 2
Z
RN
|x|−b|u(t, x)|p+2dx
− 2p p+ 2
Z
RN
|x|−b|u(t, x)|p+2(4ϕR−N)dx.
(3.11)
The second term can be estimated as follows:
− 2p
p+ 2 Z
RN
(∆ϕR−N)|x|−b|u(t, x)|p+2dx
≤C Z
|x|≥R
|x|−b|u(t, x)|p+2dx
≤CR−b Z
|x|≥R
|u(t, x)|p+2dx
≤CR−bku(t)kp+2−L2(|x|≥R)N p2s ku(t)kN p2s
LN−2s2N (|x|≥R)
≤CR−bku(t)kp+2−L2(|x|≥R)N p2s ku(t)kHN p2ss
≤CC
N p 2s
1 R−bku(t)kp+2−
N p 2s
L2(|x|≥R).
(3.12)
Thus, we have
− 2p p+ 2
Z
RN
|x|−b|u(t, x)|p+24ϕRdx
≤ −2pN p+ 2
Z
RN
|x|−b|u(t, x)|p+2dx+CC
N p 2s
1 R−bku(t)kp+2−
N p 2s
L2(|x|≥R).
(3.13)
For the last term in (3.8), we have
− 4b p+ 2
Z
RN
(x· ∇ϕR)|x|−b−2|u(t, x)|p+2dx
=− 4b p+ 2
Z
RN
|x|−b|u(t, x)|p+2dx
− 4b p+ 2
Z
|x|≥R
(x· ∇ϕR(r)
|x|2 −1)|x|−b|u(t, x)|p+2dx . By the similar method as (3.12), we deduce
− 4b
p+ 2 Z
|x|≥R
(x· ∇ϕR(r)
|x|2 −1)|x|−b|u(t, x)|p+2dx
≤CC
N p 2s
1 R−bku(t)kp+2−
N p 2s
L2(|x|≥R).
(3.14)
Collecting (3.9)-(3.14), we obtain d
dtMϕR[u(t)]≤4sk(−∆)s/2u(t)k2L2−2N p+ 4b p+ 2
Z
RN
|x|−b|u(t, x)|p+2dx +CR−2sku(t)k2L2(|x|≥R)+CC
N p 2s
1 R−bku(t)kp+2−
N p 2s
L2(|x|≥R).
(3.15)
By (3.5), we see that for anyη >0 and anyR >1, there existsC >0 independent ofRand C1 such that for anyt∈[0, T0] with T0= CCηR2
1
, d
dtMϕR[u(t)]≤4K(u(t)) +CR−2s(η+oR(1))2+CC
N p 2s
1 R−b(η+oR(1))p+2−N p2s
≤ −4δ+CR−2s(η2+oR(1)) +CC
N p 2s
1 R−b(ηp+2−N p2s +oR(1)).
We first chooseη >0 small enough andR >1 large enough so that d
dtMϕR[u(t)]≤ −δ <0, (3.16) for anyt∈[0, T0] withT0= CCηR2
1
. Note thatη >0 is fixed, so we can chooseR >1 large enough so thatT0is as large as we want. By (3.16), it follows that
MϕR[u(t)]≤ −ct,
for all t ∈ [t0, T0] with some sufficiently large t0 ∈ [0, T0]. The constant c >
0 depends only on δ. On the other hand, we deduce from Lemma 2.3 and the conservation of mass that
|MϕR[u(t)]| ≤C(ϕR)
k|∇|1/2u(t)k2L2+ku(t)kL2k|∇|1/2u(t)kL2
≤C(ϕR)
k|∇|1/2u(t)k2L2+ku(t)k2L2
≤C(ϕR)
k|∇|1/2u(t)k2L2+ 1 ,
for everyt∈[0,+∞). By interpolating betweenL2and ˙Hs, we obtain ct≤ −MϕR[u(t)] =|MϕR[u(t)]| ≤C(ϕR)
k(−∆)s/2u(t)kL1s2+ 1 ,
for anyt∈[t0, T0]. This implies that
k(−∆)s/2u(t)kL2 ≥Cts, (3.17) for all t ∈ [t1, T0] with some sufficiently large t1 ∈ [t0, T0]. Taking t close to T0= CCηR2
1
, we see thatk(−∆)s/2u(t)kL2 → ∞as R→ ∞, which contradicts (3.4).
The proof is complete.
Applying Lemma 3.1, we can prove blow-up criteria for (1.2).
Proof of Theorem 1.1. We only check that (3.1) holds under the assumptions of this Theorem. In the L2-critical case, i.e., sc = 0. The blow-up condition (1.11) implies thatku0kL2 <kQkL2 and ku0kL2 >kQkL2, which is impossible. Thus, for sc= 0 the only admissible condition isE(u0)<0. It follows from the conservation of energy andp=4s−2bN that
K(u(t)) =sku(t)k2H˙s−N p+ 2b 2p+ 4
Z
RN
|x|−b|u(t, x)|p+2dx
=2sE(u(t)) +4s−N p−2b 2p+ 4
Z
RN
|x|−b|u(t, x)|p+2dx
=2sE(u0),
for allt∈[0, T∗). Hence, whenE(u0)<0, (3.1) follows with δ=−2sE(u0).
Next, we consider the caseE(u0)>0. The assumption (1.11) implies E(u0)ku0k2σL2 < E(Q)kQk2σL2,
k(−∆)s/2u0kL2ku0kσL2 >k(−∆)s/2QkL2kQkσL2, (3.18) where
σ:=s−sc
sc
=2sp−N p+ 4s−2b N p+ 2b−4s .
We notice that the sharp constant in Gagliardo-Nirenberg inequality (2.1) can be written as
Copt= R
RN|x|−b|Q(x)|p+2dx kQk
N p+2b 2s
H˙s kQkp+2−
N p+2b 2s
L2
, (3.19)
which, by (2.2), can be rewritten as Copt= 2s(p+ 2)
N p+ 2b
1
(kQkH˙skQkσL2)N p+2b−4s2s . (3.20) It easily follows that
E(Q)kQk2σL2 =N p+ 2b−4s
2(N p+ 2b) (kQkH˙skQkσL2)2. (3.21) Multiplying both sides ofE(u(t)) byku(t)k2σL2, we deduce from the sharp Gagliardo- Nirenberg inequality (2.1) that
E(u(t))ku(t)k2σL2 =1
2ku(t)k2H˙sku(t)k2σL2− 1 p+ 2
Z
RN
|x|−b|u(t, x)|p+2dxku(t)k2σL2
≥1
2(ku(t)kH˙sku(t)kσL2)2− Copt
p+ 2(ku(t)kH˙sku(t)kσL2)N p+2b2s
=f(ku(t)kH˙sku(t)kσL2),
where f(x) := 12x2−Cp+2optxN p+2b2s . It is easy to see thatf is increasing on (0, x0) and decreasing on (x0,∞), where
x0= 2sp+ 4s Copt(N p+ 2b)
N p+2b−4s2s
=kQkH˙skQkσL2,
where the last equality follows from (3.20). It follows from (3.20) and (3.21) that f(kQkH˙skQkσL2) =E(Q)kQk2σL2.
Thus the conservation of mass and energy together with the first condition in (1.11) imply
f(ku(t)kH˙sku(t)kσL2)≤E(u(t))ku(t)k2σL2 =E(u0)ku0k2σL2
< E(Q)kQk2σL2 =f(kQkH˙skQkσL2),
for allt∈[0, T∗). Using the second condition (1.11), the continuity argument shows that
ku(t)kH˙sku(t)kσL2 >kQkH˙skQkσL2 (3.22) for any t∈[0, T∗). On the other hand, sinceE(u0)ku0k2σL2 < E(Q)kQk2σL2, we pick η >0 small enough so that
E(u0)ku0k2σL2 ≤(1−η)E(Q)kQk2σL2.
Thus, by the conservation of energy, (3.21) and (3.22), we have K(u(t))ku(t)k2σL2 = N p+ 2b
2 E(u(t))ku(t)k2σL2−N p+ 2b−4s
4 ku(t)k2H˙sku(t)k2σL2
= N p+ 2b
2 E(u0)ku0k2σL2−N p+ 2b−4s
4 (ku(t)kH˙sku(t)kσL2)2
≤ N p+ 2b
2 (1−η)E(Q)kQk2σL2−N p+ 2b−4s
4 (kQkH˙skQkσL2)2
=−ηN p+ 2b
2 E(Q)kQk2σL2,
for all t ∈ [0, T∗). This implies (3.1) with δ = ηN p+2b2 E(Q)kQk2σL2. Thus, the solutionu(t) of (1.2) blows up in finite or infinite time. This completes the proof.
4. Strong instability
In this section, we apply the blow-up criteria in Theorem 1.1 to prove Theorem 1.3.
Proof of Theorem 1.3. We divide the proof into two cases: (1)p= 4s−2bN and (2)
4s−2b
N < p < 4s−2bN−2s.
Case (1) p = 4s−2bN . Firstly, we deduce from Pohozaev’s identities (2.2) that E(Qω) = 0, where Qω is the ground state solution of (1.13). Thus, if we can construct initial datau0,n such thatE(u0,n)<0 andu0,n→QωinHs, asn→ ∞, then the corresponding solutionun blows up in finite or infinite time by applying Theorem 1.1. This implies that the standing waveu(t, x) =eiωtQω(x) is unstable.
Let {cn} ⊆C be such that |cn| >1 and limn→∞|cn| = 1, and{λn} ⊆R+ be such that limn→∞λn= 1. We take the initial data
u0,n(x) :=cnλN/2n Qω(λnx).
Then, we have
n→∞lim ku0,nkL2= lim
n→∞|cn|kQωkL2 =kQωkL2,
n→∞lim ku0,nkH˙s = lim
n→∞|cn|λsnkQωkH˙s =kQωkH˙s.
Thus, from Brezis-Lieb’s lemma we deduce thatu0,n→Qω inHsas n→ ∞.
On the other hand, from Pohozaev’s identities (2.2) we deduce that E(u0,n) =1
2ku0,nk2H˙s− 1 p+ 2
Z
RN
|x|−b|u0,n(x)|p+2dx
=|cn|2λ2sn
2 kQωk2H˙s−|cn|p+2λb+
N p
n 2
p+ 2 Z
RN
|x|−b|Qω(x)|p+2dx
=(|cn|2− |cn|p+2)λ2sn
2 kQωk2H˙s<0.
Applying Theorem 1.1, the solution un of (1.2) with initial datau0,n blows up in finite time.
